Counter-flow resistivity We first show that the counter-flow transport is insulating when \(D<D_c\)  and metallic when \(D>D_c\) . Thus the displacement field tunes an counter-flow insulator to metal transition. The counter-flow resistivity is defined as \(\rho _{s;xx}=\frac{E^s_x}{J^s_x}\) , where \(\vec{E}=\vec{E}^t-\vec{E}^b\)  and \(\vec{J}^s=\frac{1}{2}(\vec{J}^t-\vec{J}^b)\) . As shown in Eq. REF , the counter-flow resistivity \(\rho _{s}=\rho _t+\rho _b\) , where \(\rho _t,\rho _b\)  are the resistivities of the fermionic partons \(f_{t\sigma }\)  and \(f_{b\sigma }\) . When \(D<D_c\) , \(\rho _t=\infty \)  and hence \(\rho _s=\infty \) . When \(D>D_c\) , we expect that \(\rho _t=A_t T^{\alpha }\)  and \(\rho _b=A_b T^{\alpha }\) . Therefore the counter-flow resistivity is metallic \(\rho _s(T)=T^{\alpha }\) . The exponent \(\alpha \)  on the temperature dependence is likely deviating from the Fermi liquid result \(\alpha =2\)  due to the coupling to the gauge field \(a_\mu \)[1]}, [2]}, [3]}, [4]}, [5]}, though the exact value of \(\alpha \)  is still not well established theoretically. We will not try to solve this problem in this paper. Instead, we propose to measure the exponent \(\alpha \)  through counter-flow transport in experiments.
